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A program for the foundations of mathematics initiated by D. Hilbert. The aim of this program was to prove the [[Consistency|consistency]] of mathematics by precise mathematical means. Hilbert's program envisaged making precise the concept of a proof, so that these latter could become the object of a mathematical theory — [[Proof theory|proof theory]].

In order to make it possible to study proofs precisely, one gives them a single precisely-defined form. This is carried out with the aid of a formalization (cf. [[Formalization method|Formalization method]]) of the theory: the assertions of the theory are changed to finite sequences of definite signs, and the logical methods of inference — to formal rules for generating new formally-represented statements from those already proved. Thus, a mathematical theory is turned into a [[Formal system|formal system]].

Despite the fact that Hilbert's program did not achieve its ultimate goal (it could not because of the [[Gödel incompleteness theorem|Gödel incompleteness theorem]], [[#References|[3]]]), the research done within this program has had great significance for the development of many areas of mathematical logic.

The term  "formalism"  is often used as a synonym for a formal system, and generally for the notion of a calculus enabling one to change operations with objects to operations with symbols corresponding to them. In the philosophy of mathematics formalism means a view of the nature of mathematics according to which mathematics is characterized by its methods rather than by the objects it studies; its objects have no meaning other than the one derived from their formal definition (a possible  "underlying nature"  is regarded as irrelevant).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundlagen der Geometrie" , Springer  (1913)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Gentzen,   "Collected papers" , North-Holland  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Gödel,   "Ueber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I"  ''Monatsh. Math. Physik'' , '''38'''  (1931)  pp. 178–198</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. Novikov,   "Elements of mathematical logic" , Oliver &amp; Boyd  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.C. Kleene,   "Mathematical logic" , Wiley  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H.B. Curry,   "Foundations of mathematical logic" , McGraw-Hill  (1963)</TD></TR></table>